2.
Dirk Haagmans, Tonal Function: Harmony, Scales, and Intervals, Book 1 (New York:
J. Fischer & Bro., 1916). John Comfort Fillmore's translation of Riemann's 1882
lecture, "Die Natur der Harmonik" (in New Lessons in Harmony [Philadelphia:
Presser, 1887]) should be noted as the first published appearance of Riemannian
ideas in North America. Also noteworthy is the persistent, sharp criticism of
Riemann's work by Bernhard Ziehn, a German expatriate living in Chicago whose
writings were followed in the German-American community as well as in Germany
itself.

7. To be sure, the central place of the major third in
Riemann's thought, an interval he understands as primarily a harmonic one, is
the starting point for any reconstruction of Riemann's theory-building method.
But yet there are (and could have been then as well) other entry points into
dualistic systems besides the triad.

8. Rehding's extract leaves out the positive attributes of
the major mode, quite understandable given the direction of his argument at that
point. His redaction of the second and third sentences is: "The major mode�is
quite unsuited" etc. Also, in connection with Rehding's description of the
passage as "added," it is appropriate to note that the passage was in the first
German edition (1863) of Helmholtz's treatise (p. 463) and was not an addition
to subsequent editions.

9. I sketch this situation in Chapter 7 of Harmonic
Function in Chromatic Music (Chicago: University of Chicago Press, 1994).

10. On the other hand, dualism did seem to engender
interest in symmetrical musical structures even among opponents of Riemann's
dualism. See David W. Bernstein, "Symmetry and Symmetrical Inversion in
Turn-of-the-Century Theory and Practice," in Music Theory and the Exploration of
the Past, ed. Christopher Hatch and David W. Bernstein (Chicago: University of
Chicago Press, 1993), pp. 377-407.

11. Helmholtz's translator, Alexander Ellis, notes that
meantone temperament may have played a significant role here. The minor third of
the tonic triad was quite sharp and hence "much rougher" than just, whereas the
major third was just or nearly so. Helmholtz, 301, note �.

13. A Riemannian rebuttal of Helmholtz's arguments would,
I am quite sure, involve evidence of composers treating the two modes in some
equal fashion, and exhibit A would be the Well-Tempered Clavier. Here, the place
of Bach and of his great minor-mode works (e.g., the B-minor Mass) in the
reception history of German music would be critical.

Before leaving the subject of Riemann and his predecessors, I should correct
an apparent misreading of Moritz Hauptmann's minor-triad structures and
explanations, which Robert Wason caught and pointed out to me. While not a
full-gospel dualist, Hauptmann consistently thought about musical structures in
dualistic, oppositional ways. The inversional relationship between major and
minor triads was thus of great interest to him. In adapting his dialectical
system (succinctly described by Rehding on p. 24) to the minor triad, his first
impulse was to invert the representation, so that the starting point, I, was
reckoned from the upper note of the fifth, like so:

Hauptmann, p. 17,
figure no. 1

II

--

I

F

ab

C

III

--I

This analysis of the minor triad, shortened to II-III-I, is used throughout the
rest of his book and seems to be entirely satisfactory. However, through some
unexplained calculus, Hauptmann claims that the above structure "is the same as"
the following analysis:

Hauptmann, p. 17,
figure no. 2
= Rehding, Ex. 1.4a

I

--

II

F

ab

C

I

--III

I will avoid a detour into Hauptmann's opposition of "determine" and "is
determined," which seems to be the way of balancing the equation of first and
second figures, and instead point out that the two figures claim very different
things. The second, in fact, is consonant with a "co-generation" theory of
minor, in which the chord appears to have two "roots": in this case, F and ab.
Curiously, it is this figure that appears first in Rehding's discussion of the "dialectial" minor triad--as an inverted major.

The following figure then appears in Rehding's book as an illustration of
Hauptmann's "alternative explanation of the minor triad, bottom-up but no longer
dialectical" (25):

Rehding, Ex. 1.4b

I

--

II

F

ab

C

I--

III

This representation is found nowhere in Hauptmann, and is even at odds with
Hauptmann's basic claims of interval intelligibility (that is, there is no
intelligible relationship in the minor third F-ab.) Something has gone wrong
here, and I suspect a kind of typographical error involving Rehding's examples,
since the text commentary suits the "correct" versions of the examples much
better. Since more readers of Rehding's book may be consulting this review than
combing library stacks for a dusty copy of Hauptmann, I submit that discussing
the typo here is better than merely calling attention to it.

15. Ludwig van Beethoven, Sonaten für Klavier, kritisch
revidirt mit Fingersatz und genauer Bezeichnung der Phrasirung herausgegeben von
Dr. Hugo Riemann (Berlin: Simrock, 1885), II: 4. Translation by the author.
Schenker's response to this kind of work was his essay "Abolish the Phrasing
Slur" (The Masterwork in Music, vol 1 [1925], translated by William Drabkin
[Cambridge: Cambridge University Press, 1994]), in which he voices his
indignation that "�editors can have the audacity to incorporate their
interpretations into a work of art of which they have not the slightest
understanding" (20).